A Review on Bilevel Optimization: From Classical
to Evolutionary Approaches and Applications
Abstract—Bilevel optimization is defined as a mathematical
program, where an optimization problem contains another optimization problem as a constraint. These problems have received significant attention from the mathematical programming
community. Only limited work exists on bilevel problems using
evolutionary computation techniques; however, recently there has
been an increasing interest due to the proliferation of practical
applications and the potential of evolutionary algorithms in
tackling these problems. This paper provides a comprehensive
review on bilevel optimization from the basic principles to
solution strategies; both classical and evolutionary. A number
of potential application problems are also discussed. To offer
the readers insights on the prominent developments in the field
of bilevel optimization, we have performed an automated textanalysis of an extended list of papers published on bilevel
optimization to date. This paper should motivate evolutionary
computation researchers to pay more attention to this practical
yet challenging area.
Index Terms—Bilevel optimization, Stackelberg Games, Evolutionary Algorithms.
I. I NTRODUCTION
Many large-scale optimization and decision-making processes faced by public and private organizations are hierarchical in the sense that the realized outcome of any solution or
decision taken by the upper level authority (leader) to optimize
their goals is affected by the response of lower level entities
(follower), who will seek to optimize their own outcomes.
Figure 1 illustrates a general bilevel problem solving structure
involving interlinked optimization and decision-making tasks
at both levels. The figure shows that for any given upper
Upper level decision (xu) space
f
Parameter for lower level problem
Upper level
decision vector
Lower Level Objective
arXiv:1705.06270v1 [math.OC] 17 May 2017
Ankur Sinha, Pekka Malo, Kalyanmoy Deb
xu space
( , ) : A feasible bilevel solution
for the upper level optimization problem
Lower level
parametric
optimization
Lower level decision (xl) space
Optimal lower
level response
Fig. 1: A general sketch of a bilevel problem.
level decision vector, there is a corresponding (parametric)
lower level optimization problem to be solved that provides
the rational (optimal) response of the follower for the leader’s
decision. The leader’s decision vector is represented by xu and
the follower’s decision vector is represented by xl . An (xu ,
x∗l ) pair where x∗l is an optimal response to xu represents
a feasible solution to the upper level optimization problem
provided that it also satisfies the constraints in the problem.
Each level has its own objectives and constraints. One aspect
of bilevel problems is that it is not symmetric in terms of two
levels. The upper level decision maker usually has complete
knowledge of the lower level problem, while the lower level
decision maker only observes the decisions of the leader and
then optimizes its own strategies. Interestingly, an incomplete
knowledge about the follower’s optimization problem to the
leader may lead to bilevel optimization problems involving
uncertainties.
It is not uncommon that the objectives of generally profitseeking private agents can well be in conflict with those of
the controlling authority. What makes such bilevel problemsolving tasks highly relevant is that they are typically characterized by very large spillover effects to the economy as well
as the surrounding environment. Given the far-reaching future
impacts of the decisions, it is not surprising that the interest
towards bilevel programming has grown strong especially
among researchers and practitioners dealing with large-scale
public sector decision-making problems. For instance, farmers
often tend to overuse fertilizers to increase the productivity,
which leads to negative externalities such as pollution. In
[179] authors use a bilevel model to design policy measures
to control the overuse of fertilizers and its negative impact
on the environment. Apart from problems concerned with
environmentally sensitive decisions (such as allocation of
mining permits or controlling the use of fertilizers), there
has been widespread interest across a number of fields in
operations research. A good example is homeland security,
where bilevel as well as trilevel optimization frameworks
have been utilized in problems ranging from interdiction of
nuclear-weapons projects to defending critical infrastructure
and solving border security problems [9], [176], [33], [34].
In addition to the public sector challenges, there is abundant
research on bilevel decision-making problems in economics,
logistics, as well as diverse areas of computer science.
The research on decision-making problems with hierarchical
leader-follower structures (bilevel optimization) can be traced
to two roots. The first root is in the domain of game theory,
where Stackelberg [157] used bilevel programming to build
descriptive models of decision behavior and establish game-
theoretic equilibria. The second root is in the domain of
mathematical programming, where the problems appeared
as bilevel optimization problems containing a nested inner
optimization problem as a constraint of an outer optimization
problem [30]. Since then a substantial body of mathematical
literature on bilevel optimization has emerged. Given that the
hierarchical optimization structure may introduce difficulties
such as non-convexity and disconnectedness even for simpler
instances of bilevel optimization, the problems have turned out
to be surprisingly difficult to handle mathematically. Bilevel
programming is known to be strongly NP-hard [80], and it has
been proven that merely evaluating a solution for optimality
is also a NP-hard task [167]. Even in the simplest case of
linear bilevel programs, where the lower level problem has a
unique optimal solution for all the parameters, it is not likely
to find a polynomial algorithm that is capable of solving the
linear bilevel program to global optimality. The proof for the
non-existence of a polynomial time algorithm for linear bilevel
problems can be found in [64].
Due to lack of well-established solution procedures, a
complex practical problem is usually modified into simpler
single level optimization task, which is solved to arrive at
a satisficing instead of an optimal solution. For the complex
bilevel problems, classical methods often fail due to real world
difficulties such as non-linearity and discreteness. Under such
circumstances, evolutionary methods can be useful tools to
offset some of these difficulties. Recent initiatives on bilevel
optimization using evolutionary algorithms suggest that a
coordinated effort on bilevel optimization by the evolutionary community could help make significant progress on this
challenging class of optimization problems (e.g., [152], [145],
[11], [57]).
Figure 2 provides a network map of different themes on
bilevel optimization that have been studied since 1950s. The
network map shows different theoretical and application topics
that have evolved under bilevel optimization. Each link in
the map connects a subtopic with a higher level topic that
are differentiated by font sizes. Subtopics connected with a
link denote an overlap. To provide a more comprehensive
overview on the past as well as recent developments in the field
of bilevel optimization, we have organized this review paper
along the three lines; theory, applications and text-analysis
of the entire bilevel literature body. First, to formalize the
notion of bilevel programming, we begin by introducing a few
central definitions and discuss the differences between optimistic and pessimistic formulations of bilevel problems. Once
the common terminology has been established, we offer an
overview on the algorithms that have been proposed for bilevel
optimization. After a brief coverage of the commonly used
classical approaches (e.g., descent methods, penalty function
methods, and trust region methods), we move on to discuss the
developments in the field of evolutionary computation, discrete
bilevel optimization and multiobjective bilevel optimization.
The method sections are followed by a review on the central
application areas. Finally, we study the research topics, and
the evolution of interest over time. The entire bilevel litera-
ture is divided into topics and a time series analysis across
each topic is performed. The text-analysis performed in this
paper is based on a recently developed non-parametric topic
model [27], [162] for analyzing unstructured information. The
technical details on the automated text-analysis approach are
provided in an appendix. The paper concludes with a brief
discussion on the directions for future research.
II. G ENERAL F ORMULATION AND D EFINITIONS
In this section, we provide a general formulation for bilevel
optimization problem. These problems contain two levels
of optimization tasks where one optimization task is nested
within the other. The outer optimization problem is commonly
referred as the leader’s (upper level) optimization problem and
the inner optimization problem is known as the follower’s
(or lower level) optimization problem. The two levels have
their own objectives and constraints. Correspondingly, there
are also two classes of decision vectors, namely, leader’s
(upper level) decision vectors and follower’s (lower level)
decision vectors. The lower level optimization is a parametric
optimization problem that is solved with respect to the lower
level decision vectors while the upper level decision vectors
act as parameters. The lower level optimization problem is a
constraint to the upper level optimization problem, such that,
only those members are considered feasible that are lower
level optimal and also satisfy the upper level constraints.
A summary of the terminologies and notations used in the
context of bilevel optimization is given in Table I.
Definition 1. For the upper-level objective function F : Rn ×
Rm → R and lower-level objective function f : Rn × Rm →
R, the bilevel problem is given by
“min”
xu ∈XU ,xl ∈XL
F (xu , xl )
subject to
xl ∈ argmin{f (xu , xl ) : gj (xu , xl ) ≤ 0, j = 1, . . . , J}
xl ∈XL
Gk (xu , xl ) ≤ 0, k = 1, . . . , K
where Gk : XU × XL → R, k = 1, . . . , K denote the upper
level constraints, and gj : XU × XL → R represent the lower
level constraints, respectively. Equality constraints may also
exist that have been avoided for brevity.
An equivalent formulation of the above problem can be
stated in terms of set-valued mapping (multi-valued function)
as follows:
Definition 2. Let Ψ : Rn ⇒ Rm be a set-valued mapping,
Ψ(xu ) = argmin{f (xu , xl ) : gj (xu , xl ) ≤ 0, j = 1, . . . , J},
xl ∈XL
which represents the constraint defined by the lower-level
optimization problem, i.e. Ψ(xu ) ⊂ XL for every xu ∈ XU .
Then the bilevel optimization problem can be expressed as a
Constraint
Convex
Mixed Integer Program
Condition
Nonlinear Global
Solution Karush-Kuhn-Tucker
Single Level Facility Location
Transport Toll
Freight
Planning
Rail
Transit
Bus
Route
Public
Passenger
Logistics
Combinatorial
Optimality
Fuzzy
Particle Swarm
Operators
Hybrid Multi-objective
Transportation
Theory
Evolutionary
Network
Policy Decision Hierarchy
Pollution Government Firm
Sustainable Tax
Genetic
Stochastic Congestion
Transport Mobile Wireless
Emmission
Environmental Economics
Bilevel
Attacker-Defender
Equilibrium
Hierarchical
Design Optimize
Stackelberg Nash
Security Hub and Spoke
Equilibrium Constraints
Intelligence Robust Design
Structure
Trilevel Adversary Autonomous
Stackelberg Game
Machine Learning
Dynamic
Electricity
Control
Power Grid
System
Design
Parameter Estimation
Transmission
Operation
Error Validation
Architecture
Computing Task
Processor
Design
Supply Chain
Pricing
Product Manufacturer
Bidding
Retailer
Commerce
Profit
Strategic
Distribution Contract
Distribute
Fig. 2: Bilevel network-map showing connections between various applications and theory since 1950s. Each connecting link
represents either a topic connected with a subtopic, or an overlap between two subtopics.
constrained optimization problem as follows:
“min”
xu ∈XU ,xl ∈XL
F (xu , xl )
subject to
xl ∈ Ψ(xu )
Gk (xu , xl ) ≤ 0, k = 1, . . . , K
where Ψ can be interpreted as a parameterized rangeconstraint for the lower-level decision vector xl .
In the above two definitions, quotes have been used while
specifying the upper level minimization problem because of
an ambiguity that arises in case of multiple lower level optimal
solutions for any given upper level decision vector. In the
presence of multiple lower level optimal solutions there is
lack of clarity at the upper level as to which optimal solution
from the lower level should be utilized. This ambiguity can be
sorted by defining different positions that may be assumed by
the leader. The two common positions that have been widely
studied are optimistic (weak) position and pessimistic (strong)
position, which we discuss next.
A. Optimistic Position
In an optimistic position, in the presence of multiple lower
level optimal solutions, the leader expects the follower to
choose that solution from the optimal set Ψo (xu ), which leads
to the best objective function value at the upper level. The
choice function of the follower in this case may be defined as
follows:
Ψo (xu ) = argmin{F (xu , xl ) : xl ∈ Ψ(xu )}
xl ∈XL
This formulation assumes some extent of cooperation between
the two players. The bilevel optimization problem under an
optimistic position has been defined below:
min
xu ∈XU ,xl ∈XL
F (xu , xl )
subject to
xl = Ψo (xu )
Gk (xu , xl ) ≤ 0, k = 1, . . . , K
Optimistic position is more tractable as compared to the
pessimistic position; therefore, most of the studies handle
TABLE I: Summary of central notations
Category
Notation(s)
Description
Decision vectors
xu ∈ XU
xl ∈ XL
Leader’s (upper level) decision variable and decision space.
Follower’s (lower level) decision variable and decision space.
Objectives
F
f
Leader’s (upper level) objective functions.
Follower’s (lower level) objective functions.
Constraints
Gk , k = 1, . . . , K
gj , j = 1, . . . , J
Leader’s (upper level) constraint functions.
Follower’s (lower level) constraint functions.
Lower level feasible region
Ω : XU ⇒ XL
Ω(xu ) = {xl : gj (xu , xl ) ≤ 0 ∀ j}, represents the lower level
feasible region for any given upper level decision vector
Constraint region (Relaxed feasible set)
Φ = gph Ω
Φ = {(xu , xl ) : Gk (xu , xl ) ≤ 0 ∀ k, gj (xu , xl ) ≤ 0 ∀ j},
represents the region satisfying both upper and lower level constraints
Lower level reaction set
Ψ : XU ⇒ XL
Ψ(xu ) = {xl : xl ∈ argmin{f (xu , xl ) : xl ∈ Ω(xu )}}, represents
xl ∈XL
the lower level optimal solution(s) for an upper level decision vector
I = gph Ψ
I = {(xu , xl ) : (xu , xl ) ∈ Φ, xl ∈ Ψ(xu )}, represents the set
of upper level decision vectors and corresponding lower level optimal
solution(s) belonging to feasible constraint region
Choice function
ψ : XU → XL
ψ(xu ) represents the solution chosen by the follower for any upper
level decision vector. It becomes important in case of multiple lower
level optimal solutions.
Optimal value function
ϕ : XU → R
Inducible region (Feasible set)
ϕ(xu ) = min {f (xu , xl ) : xl ∈ Ω(xu )} represents the minimum
xl ∈XL
lower level function value corresponding to a given upper level decision
vector.
optimistic version of the bilevel optimization problem. The optimistic formulation is guaranteed to have an optimal solutions
under reasonable assumptions of regularity and compactness
that are stated in the theorem below:
Theorem 1. If the functions F, f, Gk and gi are sufficiently
smooth, the constraint region Φ of the bilevel optimization
problem is non-empty and compact, and the MangasarianFromowitz constraint qualification holds at all points, then the
problem is guaranteed to have an optimistic bilevel optimum
provided there exists a feasible solution.
See [81], [127], [106], [107], [61], [59] for further discussion on existence of optimistic bilevel optimum and additional
results on optimality conditions.
B. Pessimistic Position
In a pessimistic position, in the presence of multiple lower
level optimal solutions, the leader optimizes for the worst
case, i.e., she assumes that the follower may choose that
solution from the optimal set which leads to the worst objective
function value at the upper level. Such a worst case choice
function of the follower may be defined as:
Ψp (xu ) = argmax{F (xu , xl ) : xl ∈ Ψ(xu )}
xl
This formulation does not assume any form of cooperation.
The bilevel optimization problem under a pessimistic position
has been defined below:
min
xu ∈XU ,xl ∈XL
F (xu , xl )
subject to
xl = Ψp (xu )
Gk (xu , xl ) ≤ 0, k = 1, . . . , K
Pessimistic position is relatively less tractable when compared
to optimistic position. In case of an optimisitc formulation
with a convex lower level problem, it is possible to reduce the
bilevel problem to single level using the variational inequality
corresponding to the lower level problem. However, such a
straightforward single level reduction is not possible in case
of a pessimistic bilevel program. This poses significant challenges in designing methodologies that can handle pessimistic
bilevel problems. For every lower level optimization problem
solved one has to keep track of that lower level optimal
solution that is worst for the upper level. This essentially
makes a pessimistic bilevel optimization a three level task.
The pessimistic formulation is guaranteed to have an optimal
solutions under stronger assumptions, as compared to the
optimistic formulation, that are given below:
Theorem 2. If the functions F, f, Gk and gi are sufficiently
smooth, the constraint region Φ of the bilevel optimization
problem is non-empty and compact, and the set-value mapping, Ψp , is lower semi-continuous for all upper level decision
vectors, then the problem is guaranteed to have a pessimistic
bilevel optimum.
For discussion on existence of pessimistic bilevel optimum
and additional results on optimality conditions, the authors
may refer to [112], [110], [61], [63], [180].
C. Example
Below we provide a simple example of a bilevel optimization problem [72] that arises in case of two firms in a Stackelberg competition. The leader has complete knowledge about
the follower’s inverse-demand function and the cost function,
and desires to maximize it’s own profits by taking into account
the actions of the follower firm. The two firms compete solely
by choosing their production levels that maximize their profits
(Πl and Πf ), and the follower acts only after observing the
actions of the leader. Formally, this model can be presented
as follows:
max
ql ,qf
s.t.
Πl = P (ql , qf )ql − Cl (ql )
qf
(3)
where P (ql , qf ) is the unit price of the goods sold, which
depends on the todal production. The assumption is that at
the optimum, all demand is satisfied. Cl (·) is the cost of
production of the leader and Cf (·) is the cost of production
of the follower. The variables in this model are the production
levels of each firm ql and qf . The leader sets its production
level first, and then the follower chooses its production level
based on the leader’s decision. This simple model assumes
homogeneity of the products manufactured by the firms.
By assuming that the firms produce and sell homogeneous
goods, we may assume a single linear price function for both
firms as an inverse demand function of the form
P (ql , qf ) = α − β(ql , qf ),
(4)
where α, β > 0 are constants. Additionally, since costs often
tend to increase with the amount of production, we assume
convex quadratic cost functions for both firms to be of the
form
C(ql ) =
C(qf ) =
δl ql2 + γl ql + cl ,
δf qf2 + γf qf + cf ,
In this section, we provide a brief overview of the classical
algorithms that have been proposed for bilevel optimization. Given the difficult nature of bilevel problems, it is
not surprising that much of the classical literature considers
bilevel problems that are mathematically well-behaved; i.e.,
contains functions that are linear, quadratic or convex. Strong
assumptions like continuous differentiability and lower semicontinuity are quite common. A significant amount of attention
has been given to linear bilevel optimization problems with
continuous [177], [20] and combinatorial [168] variables. For
more complex bilevel problems, the readers may refer [49],
[10].
A. Single-level Reduction
(1)
qf ∈ argmax{Πf = P (ql , qf )qf − Cf (qf )},(2)
ql , qf ≥ 0,
III. C LASSICAL A PPROACHES
(5)
(6)
where ci denotes the fixed costs of the respective firm, and
δi and γi are positive constants. It is possible to solve the
above model analytically, using the first order conditions of
the lower level problem to reduce it to single level, and then
using the first order conditions of the reduced problem.
The optimal level of production of the leader (ql∗ ) and the
follower (qf∗ ) in terms of the constants of the model is given
as follows:
2(β + δf )(α − γl ) − β(α − γf )
.
(7)
ql∗ =
4(β + δf )(β + δl ) − 2β 2
β 2 (α − γf )
β(α − γl ) −
α − γf
2(β + δf )
qf∗ =
−
.
(8)
2(β + δf ) 4(β + δf )(β + δl ) − 2β 2
When the lower level problem is convex and sufficiently
regular, it is possible to replace the lower level optimization
problem with its Karush-Kuhn-Tucker (KKT) conditions. The
KKT conditions appear as Lagrangian and complementarity
constraints, and reduce the overall bilevel optimization problem to a single-level constrained optimization problem. For
example, the problem in Definition 1 can be reduced to the
following form, when the convexity and regularity conditions
at the lower level are met:
min
xu ∈XU ,xl ∈XL ,λ
F (xu , xl )
subject to
Gk (xu , xl ) ≤ 0, k = 1, . . . , K,
∇xl L(xu , xl , λ) = 0,
gj (xu , xl ) ≤ 0, j = 1, . . . , J,
λj gj (xu , xl ) = 0, j = 1, . . . , J,
λj ≥ 0, j = 1, . . . , J,
where
L(xu , xl , λ) = f (xu , xl ) +
J
X
λj gj (xu , xl ).
j=1
The above formulation, though a single level optimization
task, is not necessarily simple to handle. The Lagrangian
constraints can lead to non-convexities even when suitable
convexity assumptions are made on all the objectives and
constraints in the bilevel formulation. The complementarity
condition, inherently being combinatorial, renders the singlelevel optimization problem as a mixed integer program.
Interestingly, for linear bilevel optimization problems, the
Lagrangian constraint is also linear. Therefore, the singlelevel optimization problem is a mixed integer linear program.
Approaches based on vertex enumeration [25], [46], [163], as
well as branch-and-bound (B&B) [16], [71] have been used
to solve these problems. It is noteworthy that B&B methods
constitute an exponentially slow algorithm with the number of
integer variables. But, B&B approaches have been successfully
applied to single-level reductions of linear-quadratic [17] and
quadratic-quadratic [65], [5] bilevel problems. An extended
KKT approach has also been considered [139] for handling
linear bilevel problems.
B. Descent Methods
In addition to KKT based approaches, a number of descent
methods have been proposed for solving bilevel optimization
problems. A descent direction in bilevel optimization leads
to decrease in upper level function value while keeping the
new point feasible. Given that a point is considered feasible
only if it is lower level optimal, finding the descent direction
can be quite challenging. To resolve the problem, researchers
have investigated ways to approximate the gradient of the
upper level objective [96] as well as considered formulation
of auxiliary programs [137], [167] to determine the direction
of descent.
C. Penalty Function Methods
In penalty function methods the bilevel optimization problem is handled by solving a series of unconstrained optimization problems. The unconstrained problem is generated by
adding a penalty term that measures the extent of violation of
the constraints. The penalty term often requires a parameter
and takes the value zero for feasible points and positive
(minimization) for infeasible points. For bilevel problems, the
first attempt towards using a penalized approach was made by
[1], [2]. They replaced the lower level problem by a penalized
problem; however, the bilevel hierarchy was still maintained
and the reduced problem was still difficult to solve. Later
double penalty method was introduced in [85], where both
upper and lower level objective functions were penalized. The
problem was reduced into a single level task by replacing
the penalized lower level problem with its KKT conditions,
and then solving the single level formulation by penalization.
In a number of studies, the lower level problem is directly
replaced by its KKT conditions and then a penalized approach
is used to solve the single level problem. Few studies where
penalty function approach has been used for linear bilevel
problems are [178], [113]. In [178], the authors convert the
linear bilevel program into a penalized bilinear optimization
problem, and then solve a series of bilinear problems to find
the optimum. In [113], the authors reduce the linear bilevel
program into single level using KKT conditions, and then
append the complementary slackness condition to the upper
level objective function with a penalty. The penalized problem
is then handled using a series of linear programs.
D. Trust-region Methods
In trust-region methods, the algorithms approximate a certain region of the objective function with a model function.
The region is expanded if the approximation is good, otherwise
it is contracted. The first study using trust-region method
to solve non-linear bilevel programs was presented in [109],
where the lower level problem had a convex objective function
and linear constraints. However, no upper level constraints
were considered. Later, a more general idea was proposed
in [117], where the authors locally approximate the bilevel
program with a model involving a linear program at the upper
level and a linear variational inequality at the lower level.
Trust-region and line search ideas have been combined to
approach the bilevel optimum over iterations. Similarly, in
[51], the authors approximate the bilevel program around an
iterate with a model that itself is a linear-quadratic bilevel
program. The authors propose to solve the linear-quadratic
bilevel program using a mixed integer solver after reducing
it to a single level problem using its lower level KKT conditions. Convergence is achieved by sequentially solving linearquadratic bilevel models.
Next, we discuss about evolutionary algorithms for bilevel
optimization. At this point, we would like to refer the readers
to other review papers [50], [58], [166], [91], [151] and books
[18], [141], [61], [161], [62] on bilevel optimization.
IV. E VOLUTIONARY A PPROACHES
A. Nested Methods
Nested evolutionary algorithms are a popular approach
to handle bilevel problems, where lower level optimization
problem is solved corresponding to each and every upper
level member [154]. Though effective, nested strategies are
computationally very expensive and not viable for large scale
bilevel problems. Nested methods in the area of evolutionary
algorithms have been used in primarily two ways. The first
approach has been to use an evolutionary algorithm at the
upper level and a classical algorithm at the lower level,
while the second approach has been to utilize evolutionary
algorithms at both levels. Of course, the choice between two
approaches is determined by the complexity of the lower level
optimization problem.
One of the first evolutionary algorithms for solving bilevel
optimization problems was proposed in the early 1990s.
Mathieu et al. [119] used a nested approach with genetic
algorithm at the upper level, and linear programming at the
lower level. Another nested approach was proposed in [187],
where the upper level was an evolutionary algorithm and the
lower level was solved using Frank-Wolfe algorithm (reduced
gradient method) for every upper level member. The authors
demonstrated that the idea can be effectively utilized to solve
non-convex bilevel optimization problems.
Nested particle swarm optimization (PSO) was used in [104]
to solve bilevel optimization problems. The effectiveness of the
technique was shown on a number of standard test problems
with small number of variables, but the computational expense
of the nested procedure was not reported. A hybrid approach
was proposed in [103], where simplex-based crossover strategy
was used at the upper level, and the lower level was solved
using one of the classical approaches. The authors report the
generations and population sizes required by the algorithm that
can be used to compute the upper level function evaluations,
but they do not explicitly report the total number of lower
level function evaluations, which presumably is high.
Differential evolution (DE) based approaches have also been
used, for instance, in [192], authors used DE at the upper level
and relied on the interior point algorithm at the lower level;
similarly, in [11] authors have used DE at both levels. Authors
have also combined two different specialized evolutionary
algorithms to handle the two levels, for example, in [12]
authors use an ant colony optimization to handle the upper
level and differential evolution to handle the lower level in
a transportation routing problem. Another nested approach
utilizing ant colony algorithm for solving a bilevel model
for production-distribution planning is [37]. Scatter search
algorithms have also been employed for solving productiondistribution planning problems, for instance [40].
Through a number of approaches involving evolutionary
algorithms at one or both levels, the authors have demonstrated
the ability of their methods in solving problems that might
otherwise be difficult to handle using classical bilevel approaches. However, as already stated, most of these approaches
are practically non-scalable. With increasing number of upper
level variables, the number of lower level optimization tasks
required to be solved increases exponentially. Moreover, if the
lower level optimization problem itself is difficult to solve,
numerous instances of such a problem cannot be solved, as
required by these methods.
B. Single-level Reduction
The idea behind single-level reduction, in the context of
evolutionary algorithms, is similar to the the discussions
in Section III-A. A number of researchers in the area of
evolutionary computation have also used the KKT conditions
of the lower level to reduce the bilevel problem into a singlelevel problem. Most often, such an approach is able to solve
problems that adhere to certain regularity conditions at the
lower level because of the requirement of the KKT conditions.
However, as the reduced single-level problem is solved with
an evolutionary algorithm, usually the upper level objective
function and constraints can be more general and not adhering
to such regularities. For instance, one of the earliest papers using such an approach is by Hejazi et al. [83], who reduced the
linear bilevel problem to single-level and then used a genetic
algorithm, where chromosomes emulate the vertex points, to
solve the problem. Wang et al. [174] reduced the bilevel
problem into a single-level optimization problem using KKT
conditions, and then utilized a constraint handling scheme to
successfully solve a number of standard test problems. Their
algorithm was able to handle non-differentiability at the upper
level objective function, but not elsewhere. Later on, Wang et
al. [175] introduced an improved algorithm that was able to
handle non-convex lower level problem and performed better
than the previous approach [174]. However, the number of
function evaluations in both approaches remained quite high
(requiring function evaluations to the tune of 100,000 for 2
to 5 variable bilevel problems). In [172], the authors used
a simplex-based genetic algorithm to solve linear-quadratic
bilevel problems after reducing it to a single level task. More
recently, Jiang et al. [87] reduced the bilevel optimization
problem into a non-linear optimization problem with complementarity constraints, which is sequentially smoothed and
solved with a PSO algorithm. Along similar lines of using
lower level optimality conditions, Li [102] solved a fractional
bilevel optimization problem by utilizing optimality results of
the linear fractional lower level problem. In [170], the authors
embed the chaos search technique in PSO to solve single-level
reduced problem.
C. Metamodeling-based Methods
Metamodeling-based solution methods are commonly used
for optimization problems [171], where actual function evaluations are expensive. A meta-model or surrogate model is an
approximation of the actual model that is relatively quicker
to evaluate. Based on a small sample from the actual model,
a surrogate model can be trained and used subsequently for
optimization. Given that, for complex problems, it is hard
to approximate the entire model with a small set of sample
points, researchers often resort to iterative meta modeling
techniques, where the actual model is approximated locally
during iterations.
Bilevel optimization problems contain an inherent complexity that leads to a requirement of large number of evaluations to solve the problem. Metamodeling, when used with
population-based algorithms, offers a viable means to handle
bilevel optimization problems. In this subsection, we discuss
four ways in which metamodeling can be applied to bilevel
optimization. The discussion related to approximation of the
rational reaction set and lower level optimal value function is
a review of some recent work. However, before starting, we
refer the readers to Figure 3, which provides an understanding
of these two mappings graphically for a hypothetical bilevel
problem. We also provide a brief discussion on approximating
the bilevel problem with an auxiliary problem.
1) Reaction set mapping: One of the approaches to solve
bilevel optimization problems using evolutionary algorithms
would be through iterative approximation of the reaction
set mapping Ψ. If the Ψ-mapping (introduced in Table I)
in a bilevel optimization problem is known, it effectively
reduces the problem to single level optimization. However, this
mapping is seldom available; therefore, the approach could
be to solve the lower level problem for a few upper level
members and then utilize the lower level optimal solutions
and corresponding upper level members to generate an approximate mapping Ψ̂. It is noteworthy that approximating a
set-valued Ψ-mapping offers its own challenges and is not a
straightforward task. Assuming that an approximate mapping,
Ψ̂, can be generated, the following single level optimization
problem can be solved for a few generations of the algorithm
before deciding to further refine the reaction set.
min
xu ∈XU ,xl ∈XL
F (xu , xl )
subject to
xl ∈ Ψ̂(xu )
Gk (xu , xl ) ≤ 0, k = 1, . . . , K
Evolutionary algorithms that rely on this idea to solve bilevel
optimization problems are [145], [150], [146], [13]. In [145],
f (x(3)
u ,x l )
ϕ (x u )
f ( xu , xl )
XU
f (x(2)
u ,x l)
XU
f (x(1)
u ,x l )
x(3)
u
x(3)
u
x (2)
u
x (2)
u
x(1)
u
C
(1)
xu
Inset 3
B
A
Inset 1
Inset 1 provides the relationship of the lower level function with
respect to the upper and lower level variables. The surface of the
lower level function is sliced with three planes, wherein first
upper level member has multiple lower level optimal solutions
while other members have unique lower level optimal solution.
Inset 2 provides the rational reaction set of the follower, which
maps follower’s optimal solutions with leader’s decision vectors.
The mapping is set−valued in regions that have multiple lower
level optimal solutions corresponding to leader’s decision vectors.
Inset 3 provides the follower’s optimal value function, which is
the minimum value of follower’s objective function for any given
leader’s decision vector.
XL
XU
x(3)
u
Inset 2
x(2)
u
x(1)
u
Ψ−Mapping
(1)
Ψ(x u )
(2)
Ψ(x u )
(3)
Ψ(x u )
XL
Fig. 3: Graphical representation of rational reaction set (Ψ) and lower level optimal value function (ϕ).
[146] authors have used quadratic approximation to approximate the local reaction set. This helps in saving lower
level optimization calls when the approximation for the local
reaction set is good. In case the approximations generated by
the algorithm are not acceptable, the method defaults to a
nested approach. It is noteworthy that a bilevel algorithm that
uses a surrogate model for reaction set mapping may need not
be limited to quadratic models but other models can also be
used.
2) Optimal lower level value function: Another way to
use metamodeling would be through the approximation of
the optimal value function ϕ. If the ϕ-mapping (introduced
in Table I) is known, the bilevel problem can once again be
reduced to single level optimization problem as follows [186],
min
xu ∈XU ,xl ∈XL
F (xu , xl )
subject to
f (xu , xl ) ≤ ϕ(xu )
gj (xu , xl ) ≤ 0, j = 1, . . . , J
Gk (xu , xl ) ≤ 0, k = 1, . . . , K.
However, since the value function is seldom known, one
can attempt to approximate this function using metamodeling
techniques. The optimal value function is a single-valued
mapping; therefore, approximating this function avoids the
complexities associated with set-valued mapping. As described
previously, an approximate mapping ϕ̂, can be generated with
the population members of an evolutionary algorithm and the
following single level optimization problem can be solved with
refinements at every few generations.
min
xu ∈XU ,xl ∈XL
F (xu , xl )
subject to
f (xu , xl ) ≤ ϕ̂(xu )
gj (xu , xl ) ≤ 0, j = 1, . . . , J
Gk (xu , xl ) ≤ 0, k = 1, . . . , K.
An evolutionary approach that relies on this idea can be found
in [149], [144].
3) Bypassing lower level problem: Another way to use a
meta-model in bilevel optimization would be completely bypass the lower level problem, as follows:
min
xu ∈XU
F̂ (xu )
subject to
Ĝk (xu ) ≤ 0, k = 1, . . . , K.
Given that the optimal xl are essentially a function of xu , it
is possible to construct a single level problem by ignoring xl
completely. However, the landscape for such a single level
problem can be highly non-convex, disconnected and nondifferentiable. Advanced metamodeling techniques might be
required to use this approach, which may be beneficial for
certain classes of bilevel problems. A training set for the
metamodel can be constructed by solving few lower level
problems for different xu . Both upper level objective F and
constraint set (Gk ) can then be meta-modeled using xu alone.
Given the complex structure of such a single-level problem, it
might be sensible to create such an approximation locally. We
are currently pursuing such an approach using artificial neural
network as the meta-modeling approach.
4) Auxiliary bilevel meta-model: Building up on the trustregion methods for solving bilevel optimization problems, it is
possible to utilize the population members in an evolutionary
algorithm to formulate auxiliary bilevel problem(s). The auxiliary bilevel problem(s) may be simple enough to be solved
using faster specialized techniques. The population members
could then be updated based on the obtained auxiliary solution(s). For the moment, there does not exist any evolutionary
algorithm based on this idea, but it may be an interesting
direction to pursue in the future.
V. D ISCRETE B ILEVEL O PTIMIZATION
In this section, we would like to discuss the contributions
made towards solving discrete bilevel optimization problems.
The formulation of the bilevel problem remains the same as
described in Definitions 1 and 2, along with one or more
variables at either of the levels being discrete. Presence of
discrete variables can pose a variety of challenges depending
upon, whether the discrete variables are present at upper level,
lower level or both levels. In the classical literature, branchand-bound and branch-and-cut are some of the commonly used
techniques to handle discreteness in variables. Most of the
work on discrete bilevel optimization employs an extension
of these ideas from single-level optimization. To highlight the
kind of complexities induced in the presence of discrete variables, we consider a simple linear bilevel problem described
in [169], [18] to show how the inducible region changes based
on the upper, lower or both level variables being discrete or
continuous.
Consider the following lower level optimization problem
which we use for identifying the inducible region of a bilevel
problem:
min xl
xl
subject to
xu + xl ≤ 2,
−xu + xl ≤ 2,
5xu − 4xl ≤ 10, −5xu − 4xl ≤ 10.
For the above lower level problem there are four possible
scenarios based on the variables being continuous or discrete:
1) Continuous-Continuous bilevel program: Consider xu ∈
R and xl ∈ R
2) Discrete-Continuous bilevel program: Consider xu ∈ Z
and xl ∈ R
Dicrete−Continuous
Continous−Continuous
xl
xl
xu
xu
Continuous−Discrete
Discrete−Discrete
xl
xl
xu
xu
Fig. 4: Inducible region in different cases when the upper and
lower level variables belong to continuous or discrete sets.
3) Discrete-Discrete bilevel program: Consider xu ∈ Z and
xl ∈ Z
4) Continuous-Discrete bilevel program: Consider xu ∈ R
and xl ∈ Z
For each scenario the inducible region can be very different
that has been shown through Figure 4. The figure clearly
demonstrates how a discrete variable at any level of the
problem can lead to a disconnected search space. Of course
there could also be situations where each level has a mix of
continuous and discrete variables.
A. Discrete Bilevel Optimization Survey
One of the early works on discrete bilevel optimization was
by Vicente et al. (1996) [169], which focused on discrete linear
bilevel programs, and analyzed the properties and existence
of the optimal solution for different kinds of discretizations
arising from the upper and lower level variables. The authors
have shown in the paper that certain compactness conditions
guarantee the existence of optimal solution in continouscontinuous linear bilevel programs, discrete-continuous linear
bilevel programs and discrete-discrete linear bilevel programs.
The conditions are equivalent to stating that the inducible
region is non-empty. However, the existence conditions in
the case of continuous-discrete linear bilevel programs are
not straightforward. For instance, the inducible region for the
continuous-discrete linear bilevel problem in Figure 4 is a noncompact set that may lead to non-existence of a bilevel optimal
solution even when the inducible region is non-empty.
A few studies preceded the study by Vicente et al. (1996)
[169]. For instance, in [124] the authors solved mixed integer
linear bilevel problems. The authors pointed out the difficulties
involved in fathoming while solving mixed integer bilevel
problems using traditional branch-and-bound techniques. Certain fathoming rules used in case of mixed integer linear
programming, like fathoming when the relaxed subproblem is
worse than the value of the incumbent or fathoming when the
solution of the relaxed subproblem is feasible for the mixed
integer problem, are not directy applicable to mixed integer
linear bilevel problems. Therefore, the authors proposed a
branch-and-bound approach involving stricter fathoming conditions. However, the algorithm has a nested structure and
is not scalable beyond few integer variables, and to counter
which the authors also proposed some heuristics. This study
was followed by [19], where the same authors solved discrete
linear bilevel programs involving only binary variables using
an implicit enumeration scheme. In this approach, the authors
place a cut, similar to the one used by Bialas and Karwan
[26] (for the continuous linear bilevel program), seeking incremental improvements in the upper level objective function.
A cutting plane method utlizing the Chvátal-Gomory cut
for the continuous-discrete bilevel program was proposed in
[60]. Benders-decomposition based techniques have also been
employed to solve bilevel problems with mixed integers at the
upper level and continuous linear programs at the lower level.
The original problem is decomposed into a master and a slave
problem. Fixing the integer values converts the slave problem
into a bilevel linear program which is solved by KKT based
reduction techniques, and the solution to the slave is utilized
to create a cut for the master problem. The algorithm switches
between master and slave problems until the optimality criteria
is met. Certain studies in this direction are [136], [70], [41].
Despite the attempts made towards algorithm development
for discrete bilevel programs, the research is still open for new
methods and ideas as none of the proposed techniques would
scale well for problems with larger number of variables. Apart
from a few nested approaches and KKT-based single level
reduction approaches, to our best knowledge, there does not
exist any algorithmic study involving evolutionary algorithms
for mixed integer bilevel problems that attempt to solve
the problem efficiently by utilizing its properties. However,
given that the evolutionary approaches are, in particular,
potent for handling difficulties such as discreteness and nondifferentiabilities they offer a significant scope for solving
discrete bilevel optimization problems. Some attempts towards
mixed integer bilevel optimization using evolutionary methods
are [82], [14], [121], [101], [4], [38], [79], [44].
There is no dearth of application problems involving discrete variables. While mixed integer bilevel problems are ubiquitous, even combinatorial bilevel programs find innumerable
applications in the areas of network design, facility location,
hub-and-spoke networks etc. In these areas, these problems are
commonly studied in the context of interdiction, protection,
robust design, competition and supply chain management,
among others. Some of the related applications are highlighted
in Section VII.
VI. M ULTIOBJECTIVE B ILEVEL O PTIMIZATION
In many of the practical problems, a leader and/or the
follower might face multiple objectives. This gives rise to
multiobjective bilevel optimization problems that we define
below.
Definition 3. For the upper-level objective function F : Rn ×
Rm → Rp and lower-level objective function f : Rn × Rm →
Rq , the multi-objective bilevel problem is given by
“min”
xu ∈XU ,xl ∈XL
F (xu , xl ) = (F1 (xu , xl ), . . . , Fp (xu , xl ))
subject to
xl ∈ argmin{f (xu , xl ) = (f1 (xu , xl ), . . . , fq (xu , xl )) :
xl ∈XL
gj (xu , xl ) ≤ 0, j = 1, . . . , J}
Gk (xu , xl ) ≤ 0, k = 1, . . . , K
where Gk : XU × XL → R, k = 1, . . . , K denote the upper
level constraints, and gj : XU × XL → R represent the lower
level constraints, respectively.
The set-valued mapping in this case can be defined as
follows and an equivalent definition can be written as in the
single-objective case:
Ψ(xu ) = argmin{f (xu , xl ) = (f1 (xu , xl ), . . . , fq (xu , xl )) :
xl ∈XL
gj (xu , xl ) ≤ 0, j = 1, . . . , J},
A. Optimistic vs Pessimistic
The optimistic or pessimistic position becomes more prominent in multiobjective bilevel optimization. In the presence of
multiple objectives at the lower level, the set-valued mapping
Ψ(·) normally represents a set of Pareto-optimal solutions
corresponding to any given xu , which we refer as follower’s
Pareto-optimal frontier. A solution to the overall problem (with
optimistic or pessimistic position) is expected to produce a
trade-off frontier for the leader that we refer as the leader’s
Pareto-optimal frontier. From the perspective of the leader, it
becomes important that what kind of position she seeks to
take while solving the problem, as it determines that which
solution(s) from the lower level frontier should be considered
at the upper level.
Though optimistic positions have commonly been studied
in classical [67] and evolutionary [57] literature in the context
of multiobjective bilevel optimization; it is far from realism
to expect that the follower will cooperate to an extent that
she chooses any point from her Pareto-optimal frontier that
is most suitable for the leader. This relies on the assumption
that the follower is indifferent to the entire set of optimal
solutions, and therefore decides to cooperate. The situation
was entirely different in the single-objective case, where, in
case of multiple optimal solutions, all the solutions offered
an equal value to the follower. However, this can not be
assumed in the multiobjective case. Solution to the optimistic
formulation in multiobjective bilevel optimization leads to the
best possible Pareto-optimal frontier that can be achieved by
Follower’s
problem
(1)
f2
for x u
Al
f2
0
Follower’s
problem
(2)
for x u
0
f1
Bl
1
0
Pessimistic
PO front
Au
0
f1
F2
Follower’s
problem
The mining company has a sole objective of maximizing its
profit under the constraints set by the government. In this
scenario, the government would like to have a tax structure
such that it is able to maximize its own revenues in addition
to being able to restrain the mining company from causing
extensive damage to the environment. It is possible for the
leader to optimally regulate the problem in its favour, provided
that it has complete knowledge of the follower’s strategies.
The hierarchical optimization problem in this case can be
formulated as follows:
(3)
for x u
f2
Bu
max
τ,q
0
Optimistic
PO front
Cl
0
f1
s.t.
Cu
q ≥ 0, τ ≥ 0.
0
0
F1
F(q, τ ) = (R, −D)
(
)
π(q) = p(q)q − c(q) − R
q ∈ argmax
π(q) ≥ 0
q
(9)
(10)
(11)
1
Fig. 5: Leader’s Pareto-optimal (PO) frontiers for optimistic
and pessimistic positions. Few follower’s Pareto-optimal (PO)
frontiers are shown (in insets) along with their representations
in the leader’s objective space.
the leader. Similarly, solution to the pessimistic formulation
leads to the worst possible Pareto-optimal frontier at the upper
level.
If the value function or the choice function of the follower is
known to the leader, it provides an information as to what kind
of trade-off is preferred by the follower. A knowledge of such
a function effectively, casually speaking, reduces the lower
level optimization problem into a single-objective optimization
task, where the value function may be directly optimized. The
leader’s Pareto-optimal frontier for such intermediate positions
lies between the optimistic and the pessimistic frontiers. Figure
5 shows the optimistic and pessimistic frontiers for a hypothetical multiobjective bilevel problem with two objectives at
upper and lower levels. Follower’s frontier corresponding to
x1u , x2u and x3u , and her decisions Al , Bl and Cl are shown in
the insets. The corresponding representations of the follower’s
frontier and decisions (Au , Bu and Cu ) in the leader’s space
are also shown.
B. Example
Below, we provide a bilevel optimization problem involving
design of a tax policy [153]. The upper level in this example
is the governemnt that wants to tax the lower level, a mining
company, based on the pollution it causes to the environment.
The government here has two objectives: the first objective is
to maximize the revenues generated by the mining project,
which may include the additional jobs, taxes, etc; and the
second objective is to minimize the harm caused to the environment as a result of mining. Obviously, there is a trade-off
between the two objectives, and the government as a decision
maker needs to choose one of the preferred trade-off solutions.
In (9), the first objective deals with the tax revenue, where
R = τ q; τ is the per unit tax imposed on the mine, and q is the
amount of metal extracted from the ore by the follower. The
second objective denotes the environmental damage caused by
the mine that the government ultimately wants to minimize.
D = kq, where k is the pollution coefficient signifying
the negative impact of extraction on the environment. The
damages are thus linear and scale proportionately with the
amount of gold extracted from the earth since a larger base of
operation implies larger environmental damage.
Equation (10) gives the profit of the mine, where p(q)q
(price function times amount of metal extracted) is the revenue
function, and c(q) is the extraction cost function followed by
the additional tax levied on the mine. The mine is most likely
to be a price taker when it comes to the price of gold and
must base its mining decisions on the possible price paid by
their customers. It would therefore be plausible to replace the
price function for gold in the above equation by a constant.
However, given the assumption that the mine can extract a
large amount of ore, and subsequently gold, at one time, it
might be possible for it to affect the price of gold slightly.
Therefore, we assume the price function to be linear with a
small slope. Extraction cost can be considered to be quadratic.
Thus, we have the following model:
max
τ,q
F (q, τ ) = (τ q, −kq)
(12)
s.t.





 π(q) =(α − βq)q−
q ∈ argmax
(δq 2 + γq + φ) − τ q


q


π(q) ≥ 0
(13)
q ≥ 0, τ ≥ 0,
(14)
where α, β, δ, γ, φ are constants, and φ represents the fixed
costs of setting up operations. The above problem can be
solved analytically by taking a weighted sum of squares of
the upper level objectives (wτ q − (1 − w)kq : w ∈ [0, 1]). The
0
−2
Objective 2 (−D)
−4
−6
−8
−10
−12
0
100
200
300
400
Objective 1 (R)
500
600
Fig. 6: Pareto-optimal frontier for the government showing the
trade-off between tax revenues and environmental pollution.
optimal solution to the above problem is given as follows:
α−γ−k
k
+
,
2
2w
w(α − γ) − (1 − w)k
q ∗ (w) =
.
4w(β + δ)
τ ∗ (w) =
(15)
(16)
We assume the parameters as α = 100, β = 1, δ = 1, γ = 1
and φ = 0. By varying the government’s preference weights
(w) in its domain, one can generate the entire Pareto-optimal
solutions for the leader. Note that a very high taxation (or
weight to the environmental objective) may lead to no production at the lower level, for instance, we find that w < 0.01
does not lead to any production. The Pareto frontier generated
using weights 0.01 ≤ w ≤ 1 has been provided in Figure 6
for the above model.
One of the first studies, utilizing an evolutionary approach
for multiobjective bilevel optimization was by Yin [187]. The
study involved multiple objectives at the upper lever, and
a single objective at the lower level. The study suggested
a nested genetic algorithm, and applied it on a transportation planning and management problem. Later Halter and
Mostaghim [77] used a particle swarm optimization (PSO)
based nested strategy to solve a multi-component chemical
system. The lower level problem in their application was
linear for which they used a specialized linear multi-objective
PSO approach. Recently, a hybrid bilevel evolutionary multiobjective optimization algorithm coupled with local search was
proposed in [57] (For earlier versions, refer [54], [143], [56],
[55]). In the paper, the authors handled non-linear as well
as discrete bilevel problems with relatively larger number of
variables. The study also provided a suite of test problems for
bilevel multi-objective optimization.
There has been some work done on decision making aspects
at upper and lower levels. For example, in [142] an optimistic version of multiobjective bilevel optimization, involving
interaction with the upper level decision maker, has been
solved. The approach leads to the most preferred point at the
upper level instead of the entire Pareto-frontier. Since multiobjective bilevel optimization is computationally expensive,
such an approach was justified as it led to enormous savings in
computational expense. Studies that have considered decision
making at the lower level include [147], [152]. In [147], the
authors have replaced the lower level with a value function that
effectively reduces the lower level problem to single-objective
optimization task. In [152], the follower’s value function is
known with uncertainty, and the authors propose a strategy
to handle such problems. Other work related to bilevel multiobjective optimization can be found in [129], [130], [108],
[135], [191].
VII. A PPLICATIONS
C. Multiobjective Bilevel Optimization Survey
There exists a significant amount of work on single objective
bilevel optimization; however, little has been done on multiobjective bilevel optimization primarily because of the computational and decision making complexities that these problems
offer. For results on optimality conditions in multiobjective
bilevel optimization, the readers may refer to [73], [185], [15].
On the methodology side, Eichfelder [66], [67] solved simple
multi-objective bilevel problems using a classical approach.
The lower level problems in these studies have been solved
using a numerical optimization technique, and the upper
level problem is handled using an adaptive exhaustive search
method. This makes the solution procedure computationally
demanding and non-scalable to large-scale problems. In another study, Shi and Xia [140] used -constraint method at
both levels of multi-objective bilevel problem to convert the
problem into an -constraint bilevel problem. The -parameter
is elicited from the decision maker, and the problem is solved
by replacing the lower level constrained optimization problem
with its KKT conditions.
Bilevel optimization commonly appears in many practical
problems. They are often encountered in the fields of economics, transportation, engineering and management, among
others. The following list will provide an insight to the readers
on the relevance of these problems to practice.
1) Toll Setting Problem: Toll-setting problem is essentially
a part of network problems. In this problem, there is an
authority that wants to optimize the tolls for a network
of roads. The authority acts as a leader and the network
users act as followers. Papers on toll-setting problem
and its multi-objective extensions can be found in [32],
[122], [52], [98], [189], [116], [90], [173], [148], [68],
[76], [92]. Bilevel optimization is quite commonly used
in network design problems. Instead of going through
specific problems, we refer the readers to a variety
of applications of bilevel optimization to the area of
network design [100], [21], [115], [184], [188], [43],
[183], [45], [74], [69], [120], [181], [134], [39].
2) Environmental Economics: Bilevel optimization commonly appears in environmental economics, where an
Fig. 7: Interest on bilevel optimization over time.
authority wants to tax an organization or individual that
is polluting the environment as a result of its operations.
Finding an optimal level of tax that offers a compromise
between revenues and pollution results in a bilevel
optimization problem with the regulator as the leader
and the polluting entity as a follower [8], [153], [28],
[29], [179].
3) Chemical Industry: In chemical industries, the chemists
often face a bilevel optimization problem where they
have to decide upon the conditions (state variables and
quantity of reactants) for the reaction to achieve optimal
output. While optimizing the output is the upper level
problem, the lower level appears as an equilibrium
condition, which is an entropy functional minimization
problem. Such applications of bilevel optimization can
be found in [156], [48], [131], [78]
4) Optimal design: Bilevel problems are very common in
structural optimization or optimal shape design. For
instance, in structural optimization one often requires
to minimize the weight or cost of a structure as an
upper level objective with the decision variables as
shape of the structure, choice of materials, amount of
material etc. The constraints at the upper level involve
bounds on displacements, stresses and contact forces
whose values are determined by solving the potential
energy minimization problem at the lower level. The
equilibrium condition in many optimal shape design
problems appears in the form of variational inequalities
which require the overall problem to formulated as a two
level task. For optimal design applications the readers
may refer to [84], [47], [22], [94], [95].
5) Defense applications: Bilevel optimization has a number
of applications in the defense sector [31], for example
attacker-defender Stackelberg games [86], [35], [138],
[126], [9], [132], [133]. Specifically, some recent applications include planning the prepositioning of defensive
missile interceptors to counter an attack threat [33],
interdicting nuclear weapons project [34], homeland
security applications [176], [111], location problems [3].
The bilevel problem, while offending, involves maxi-
Fig. 8: Interest on evolutionary bilevel optimization over time.
mizing the damage caused to the opponent by taking
into account the optimal reactions of the opponent. Conversely, while defending, the bilevel problem involves
minimizing the maximum damage that an attacker can
cause.
6) Facility Location: Facility location problems may take
the form of a Stackelberg game if a firm, while locating
its facility, decides to account the actions of its competitors. For instance, in [97] the authors study the scenario
where a firm enters a market by locating new facilities,
and its competitor reacts by adjusting the attractiveness
of its existing facilities. Another study considers location of logistics distribution centers by minimizing the
planners’ cost at the upper level and customers’ cost
at the lower level [158]. Other applications of bilevel
optimization to facility location problem may be found
in [88], [164], [159], [7], [36], [38], [128], [38], [118],
[41], [114].
7) Inverse Optimal control: Inverse optimal control problems are essentially bilevel in nature [123], [6], [89],
[160] with wide applications in robotics, computer vision, communication theory and remote sensing to name
a few. One of the major challenges in control theory is
deriving the performance index or reward function which
fits best on a given dataset. Such tasks lie in the category
of inverse optimal control theory, where one solicits the
calculation of the cause based on the given result. Such a
requirement necessitates solving a parameter estimation
problem with an optimal control problem.
8) Machine learning: Most of the machine learning and
evolutionary optimization techniques often involve a
number of parameters. A proper choice of these parameters has a substantial effect on the accuracy and
efficiency of the approach. Tuning of these parameters
is often achieved using brute force strategies, such as
grid search and random search. A bilevel formulation
of this problem allows for systematic and more efficient
search when the number of parameters are large. Some
of the approaches that have acknowledged the bilevel
nature of this problem are [23], [24], [155], [105].
9) Principal-agent problems: Principal-agent problem [99]
is a classical problem in economics, where a principal
(leader) sub-contracts a job to an agent (follower).
Given that the agent prefers to act in his own interests rather than those of the principal, it becomes
important for the principal to have an incentive scheme
that aligns the interests of the agent with the principal. Design of such contracts appears as a bilevel
optimization problem. In real life, principal-agent relationships are commonly found in doctor-patient, senior
management-lower management, employer-employee,
corporate board-shareholders and politician-voters scenarios. Studies that the readers may refer to are [165],
[75], [190], [42], [182].
Fig. 9: Topic: Methods.
VIII. I NTEREST OVER TIME
In this section, we perform a text analysis of papers on
bilevel programming (and Stackelberg games) that are indexed
in SCOPUS. To begin with, we analyze the volume of publications every year on bilevel programming since 1950s to
present, and then closely look at the themes within bilevel programming that have contributed to the growth over the years.
The themes were discovered using a non-parametric Bayesian
approach [162], which clusters the documents together based
on similarities. The documents may probabilistically belong
to multiple clusters at the same time.
Figure 7 shows how the interest on bilevel programming has
been growing at a slow pace until early 2000s and then pickedup significantly at the middle of the previous decade. The
studies on bilevel programming using evolutionary algorithms
appeared for the first time during the mid 1990s that took
another decade to pick up to the extent that almost 10% of all
studies on bilevel optimization utilize evolutionary methods.
Figure 7 shows the growth of evolutionary methods in the
context of bilevel optimization from the 1990s to present.
While the early papers on bilevel programming (pre-2000)
were mainly focused on solution methods and optimality
conditions, the growth in the post-2000 period was fueled by
papers on applications of bilevel programming.
To identify the themes that have contributed towards the
literature on bilevel optimization we used topic models. “Topic
models are algorithms for discovering the main themes that
pervade a large and otherwise unstructured collection of
documents.” [27] These models can be used to organize
unstructured collections as well as develop insights from large
text databases that made it suitable for our purposes. The
results from the topic model for the documents retrieved from
SCOPUS are given in Figures 9-24. The figures consist of
themes that have a volume at least to the order of around 1%.
Along with the identification of themes, the analysis helped
in determining the attention received by particular themes at
any point in time.
Each figure contains a word cloud in the inset that describes
the theme and volume of papers as the other inset that
describes the number of papers published on that theme over
the years. Interestingly, we observe that number of papers
Fig. 10: Topic: Optimality conditions.
Fig. 11: Topic: Classical game theory.
Fig. 12: Topic: Network design.
on classical bilevel methods and optimality conditions peaked
during 1995-2000. Since 2000 a number of bilevel applications
picked-up, for example, we see a growth in supply chain
applications, electricity transmission applications, telecommunication applications, facility location applications, railway applications and machine learning applications. Defense applications that appeared to be minimally present before 2000 show
a significant presence later. Network design, optimal design
and business applications do not show a trend but represent
the highest volume of applications on bilevel optimization.
Fig. 13: Topic: Supply chain applications.
Fig. 17: Topic: Business applications.
Fig. 14: Topic: Optimal design applications.
Fig. 18: Topic: Computer architecture and circuit design.
Fig. 15: Topic: Electricity transmission applications.
Fig. 19: Topic: Hierarchical decision making applications.
Fig. 16: Topic: Telecommunication applications.
Fig. 20: Topic: Environment applications.
IX. C ONCLUSIONS AND F UTURE R ESEARCH D IRECTIONS
A. Metamodeling-based Algorithms
In this concluding section, we will raise a few perspectives
that have not yet received much attention but may offer interesting directions for future research. Apart from metamodeling
based techniques to solve bilevel problems, we would like
to highlight the importance of being able to account for
different forms of uncertainties that are often encountered
when solving practical problems. Another interesting direction
is concerned with scalability of bilevel algorithms and ability
to leverage distributed computing platforms to handle large
scale problems.
Though we have already highlighted the importance of
metamodeling-based methods in an earlier section, we have
decided to discuss it once again because of its potential
in solving practical bilevel optimization problems. Bilevel
optimization problems inherit a number of mappings, and
any knowledge about the structure of these mappings can
simplify the solution procedure extensively. In this paper, we
have highlighted approaches that are based on approximating
the reaction set mapping and the optimal lower level value
function mapping. Knowing one of these mappings reduces
B. Multiobjective Bilevel Optimization and Decision Making
Fig. 21: Topic: Facility location applications.
Fig. 22: Topic: Railway applications.
Multiobjective bilevel optimization has received only lukewarm interest from researchers. A number of issues, like
decision interaction between the two levels and uncertainties
in decision making, remain unexplored. It might be of interest
for researchers working in the area of multi-criteria decision
making and multiobjective evolutionary optimization to explore how two levels of decision makers interact to arrive
at a compromising or an equilibrium solution in different
situations. Similarly, the notion of uncertain decision-maker’s
preferences at one or both levels has also not received enough
attention [152]. In the field of Decision Analysis, however,
plenty of research has been carried out to extend traditional
frameworks of decision making such as Expected Utility Theory [125] and Multi-attribute Utility Theory [93] to account
for uncertain preferences on, for instance, the trade-offs among
multiple decision objectives or the risk-attitude. However,
preferential uncertainty in bilevel optimization problems still
requires development of theory as well as methods to account
for decision behavior in a hierarchical setting. The approaches
that have been proposed so far are still very preliminary and
require substantial future research.
C. Bilevel Optimization under Variable Uncertainty
Fig. 23: Topic: Machine learning applications.
Another important research topic is concerned with the
inherent uncertainty of decision variables. This poses several
challenges for the existing more deterministic optimization
frameworks that may fail to find solutions that are robust and
sufficiently close to the optimal solutions. The fact that bilevel
problems have nested optimization tasks makes the search
of robust solutions substantially more challenging compared
with single-level optimization problems. A few preliminary
ideas for handling variable uncertainty have already been
suggested [53], but the required algorithm side innovations
that would make these problems accessible to practitioners
are still missing.
D. Scaling of Evolutionary Bilevel Algorithms
Fig. 24: Topic: Defense applications.
any bilevel optimization problem to a single-level optimization
problem. For solving large scale bilevel problems, one has
to exploit the structure and properties of bilevel problems
that are essentially contained in these mappings. Other ways
of utilizing metamodeling for bilevel problems that we have
discussed are: approximating the bilevel problem by bypassing
the lower level problem completely; and utilizing auxiliary
bilevel models to locally approximate a bilevel optimization
problem. Only few studies in the context of evolutionary
algorithms utilize such a strategies and offer opportunities for
future contributions.
Bilevel problems are well-known for being highly computationally intensive already before considering any types of
uncertainties. One of the promising directions for handling
larger bilevel problems could be the use of the recent distributed computing platforms such as Apache Spark project.
The programming model of Spark is quite different from the
Hadoop MapReduce framework, and it has managed to overcome many of the earlier limitations. In particular, its current
form has turned out to support the use and development of
iterative algorithms quite well. Therefore, it may be interesting
to investigate whether this novel platform will be able to offer
a helping hand for researchers and practitioners who seek to
solve bigger bilevel problems faster.
Though considerable progress has already been made during
the last few years, evolutionary bilevel optimization is still a
relatively young field with numerous opportunities for both
computational as well as theoretical innovations. The growing
availability of algorithms is also opening the field to more
applied research, and we believe that in the future we are
likely to see a considerable amount of novel applications.
X. ACKNOWLEDGMENTS
Ankur Sinha and Pekka Malo would like to acknowledge the
support provided by Liikesivistysrahasto and Helsinki School
of Economics Foundation. K. Deb acknowledges the support
from NSF Beacon Center for the study of evolution in action
at MSU under Cooperative Agreement No. DBI-0939454.
Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s) and
do not necessarily reflect the views of the National Science
Foundation.
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